All questions of Mechanical Properties of Solids for Class 11 Exam

When a soild is compressed or elongated, it's P.E increases. Potential energy increases because work has to be done against force of repulsion during compression and against force of attraction during elongation.

Given:
- The flood lamp stretches the wire by 0.18 mm
- The stress is proportional to the strain
To find:
- How much would it have stretched if the wire had the same length but twice the diameter

Let's begin by understanding the given information.

Stress is defined as the force per unit area and is denoted by the symbol σ (sigma). Mathematically, stress is given by:

σ = F / A

where F is the force applied and A is the area over which the force is applied.

Strain is defined as the change in length per unit length and is denoted by the symbol ε (epsilon). Mathematically, strain is given by:

ε = ΔL / L

where ΔL is the change in length and L is the original length.

From the given information, we know that the stress is proportional to the strain. This can be expressed mathematically as:

σ ∝ ε

or

σ = kε

where k is a constant of proportionality.

Now, let's apply this information to the problem at hand.

When the flood lamp is hung from the wire, it exerts a force on the wire which causes it to stretch. Let's assume that the original diameter of the wire is d and the original length is L.

From the given information, we know that the stress is proportional to the strain. Therefore, we can write:

σ = kε

where σ is the stress, k is a constant of proportionality, and ε is the strain.

The stress can be calculated using the formula:

σ = F / A

where F is the force applied and A is the cross-sectional area of the wire.

The force applied is the weight of the flood lamp, which can be calculated using the formula:

F = mg

where m is the mass of the flood lamp and g is the acceleration due to gravity.

The cross-sectional area of the wire can be calculated using the formula:

A = πd^2 / 4

where d is the diameter of the wire.

Therefore, we can write:

σ = (mg) / (πd^2 / 4)

The strain can be calculated using the formula:

ε = ΔL / L

where ΔL is the change in length and L is the original length.

From the given information, we know that the flood lamp stretches the wire by 0.18 mm. Therefore, we can write:

ε = 0.18 / L

Now, let's combine the equations for stress and strain:

σ = kε

σ = (mg) / (πd^2 / 4)

ε = 0.18 / L

Substituting the values of σ and ε, we get:

(mg) / (πd^2 / 4) = k (0.18 / L)

Simplifying, we get:

k = (mgL) / (0.18πd^2)

Now, let's use this value of k to calculate the change in length when the diameter of the wire is doubled.

When the diameter of the wire is doubled, the cross-sectional area of the wire becomes 4 times the original area. Therefore, the new diameter is 2d and the new cross-sectional area is:

A' = π(2d)^2 / 4 = 4πd^2

Using the same formula for stress,

Luggage

From Hooke's Law: F =kx

Tensile stress (or tension) is the stress state leading to expansion; that is, the length of a material tends to increase in the tensile direction. The volume of the material stays constant. When equal and opposite forces are applied on a body, then the stress due to this force is called tensile stress.

Given data:
Bulk modulus of glass = 37 GPa
Hydraulic pressure = 10 atm

Formula used:
Bulk modulus of elasticity (K) = (pressure * volume)/(volume change)

Calculation:
Let initial volume of the glass slab be V₀ and let the change in volume be ΔV.
Bulk modulus of elasticity (K) = (pressure * volume)/(volume change)

⇒ K = (10 atm * V₀)/(ΔV)

⇒ ΔV/V₀ = (10 atm * V₀)/K

⇒ ΔV/V₀ = (10 atm * V₀)/(37 × 10⁹ N/m²)

⇒ ΔV/V₀ = 2.7 × 10⁻⁶

Hence, the fractional change in volume of a glass slab, when subjected to a hydraulic pressure of 10 atm, is 0.0027. Therefore, option A is the correct answer.

Introduction:
Hooke's Law is a principle in physics that relates the force applied to a spring or elastic material to the resulting deformation or change in length of the material. It states that the force applied to a spring is directly proportional to the displacement or change in length of the spring.

Explanation:
The constant of proportionality in Hooke's Law is known as the modulus of elasticity or Young's modulus. It is represented by the symbol 'E' and is a measure of the stiffness or rigidity of a material. The modulus of elasticity signifies how much a material will deform when a force is applied to it.

Modulus of Elasticity:
The modulus of elasticity is a material property that describes how it responds to stress. It is defined as the ratio of stress to strain within the elastic limit of the material. In other words, it measures how much stress a material can withstand before it starts to deform permanently.

Modulus of Strain:
The modulus of strain is not a property used in Hooke's Law. Strain is the measure of deformation or change in length of a material, and the modulus of strain is not directly related to the constant of proportionality in Hooke's Law.

Elasticity of Wire:
The elasticity of a wire refers to its ability to return to its original shape after being stretched or deformed. It is related to Hooke's Law as the law describes the linear relationship between the force applied to a wire and the resulting deformation or change in length of the wire.

Modulus of Stress:
The modulus of stress is not a term used in Hooke's Law. Stress is defined as the force applied per unit area of a material, and the modulus of stress is not directly related to the constant of proportionality in Hooke's Law.

Conclusion:
In conclusion, the constant of proportionality in Hooke's Law signifies the modulus of elasticity. It is a measure of the stiffness or rigidity of a material and describes how much a material will deform when a force is applied to it. The modulus of elasticity is a fundamental property used to understand the behavior of elastic materials and is essential in various fields such as engineering and materials science.

Alternation of generations (also known as metagenesis) is the type of life cycle that occurs in those plants and algae in the Archaeplastida and the Heterokontophyta that have distinct sexual haploid and asexual diploid stages.

I section is generally used as a beam because of its high section modulus as it's most of the area is situated away from it's neutral axis hence it has high moment of inertia i.e high section modulus i.e high moment carrying capacity which is the major requirement for a good beam section.

Understanding Volume Strain
Volume strain is an important concept in mechanics and materials science that describes how a material deforms when subjected to external forces.
Definition of Volume Strain
- Volume strain is defined specifically as the ratio of the change in volume (ΔV) to the original volume (V0) of a material.
- Mathematically, it can be expressed as: Volume Strain = ΔV / V0.
Why Option B is Correct
- Change in Volume (ΔV): This represents the difference between the final volume after deformation and the initial volume before deformation.
- Original Volume (V0): This is the volume of the material before any external forces have been applied.
- Ratio Significance: By taking the ratio of the change in volume to the original volume, we obtain a dimensionless quantity that allows for comparison across different materials and conditions.
Other Options Explained
- Option A (Change in Volume V): This does not provide a comparative metric and lacks the necessary context of the original volume.
- Option C (Thrice the Original Volume): This is an arbitrary scaling that does not conform to the standard definition of volume strain.
- Option D (Twice the Original Volume): Similar to Option C, this does not reflect the true relationship defined in mechanics.
Conclusion
In conclusion, volume strain is fundamentally about understanding how a material's volume changes relative to its original volume, which is effectively captured by Option B. This definition is crucial for engineers and scientists to assess material behavior under stress.

An elastomer is a polymer with viscoelasticity (i. e., both viscosity and elasticity) and very weak intermolecular forces, and generally low Young's modulus and high failure strain compared with other materials. Elastomer rubber compounds are made from five to ten ingredients, each ingredient playing a specific role. Polymer is the main component, and determines heat and chemical resistance, as well as low- temperature performance. Reinforcing filler is used, typically carbon black, for strength properties.

Brittle materials are materials that show very little or no plastic deformation before fracture. When a tensile force is applied to a brittle material, it undergoes elastic deformation until it reaches its elastic limit. Beyond the elastic limit, the material fractures without any significant plastic deformation. Therefore, the correct answer is option 'A' - brittle materials.

Explanation:

1. Brittle materials:
- Brittle materials are characterized by their lack of ductility or ability to deform plastically.
- They have limited or no plastic range beyond their elastic limit.
- Examples of brittle materials include glass, ceramics, and some types of polymers.

2. Elastic limit:
- The elastic limit is the maximum stress or force that a material can withstand without undergoing permanent deformation.
- It is the point beyond which the material no longer returns to its original shape after the applied stress is removed.

3. Plastic deformation:
- Plastic deformation refers to the permanent change in shape or size of a material under the influence of an external force.
- In the context of this question, it specifically refers to the ability of a material to undergo significant plastic deformation before fracture.

4. Elastic materials:
- Elastic materials are those that can deform under the influence of external forces and return to their original shape when the forces are removed.
- These materials have a significant elastic range, where deformation is reversible, before reaching their elastic limit.
- Examples of elastic materials include rubber and certain types of polymers.

5. Elastomers:
- Elastomers are a specific type of elastic material that exhibits high elasticity and can undergo large elastic deformations.
- They can stretch and recover their original shape due to their unique molecular structure.
- Examples of elastomers include natural rubber and synthetic rubber.

In summary, materials that show very small plastic range beyond their elastic limit are classified as brittle materials. These materials fracture without undergoing significant plastic deformation.